First slide

Rectilinear Motion

Question

A body starts from rest with uniform acceleration. If its velocity after n second is v, then its displacement in the last two seconds is

Moderate

A

$\frac{2 v (n + 1)}{n}$
B

$\frac{v (n + 1)}{n}$
C

$\frac{v (n - 1)}{n}$
D

$\frac{2 v (n - 1)}{n}$

Solution

$∵ v = 0 + n a \Rightarrow a = v / n$
$Now, distance travelled in n sec α \Rightarrow S_{n} = \frac{1}{2} a n^{2} and distance travelledin (n - 2) \sec \Rightarrow S_{n - 2} = \frac{1}{2} a (n - 2)^{2}$
$∴$ Distance travelled in last two seconds,
$\begin{array}{l} = S_{n} - S_{n - 2} = \frac{1}{2} a n^{2} - \frac{1}{2} a (n - 2)^{2} \\ = \frac{a}{2} [n^{2} - (n - 2)^{2}] = \frac{a}{2} [n + (n - 2)] [n - (n - 2)] \\ = a (2 n - 2) = \frac{v}{n} (2 n - 2) = \frac{2 v (n - 1)}{n} \end{array}$

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